1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Order of variables in a Jacobian?

  1. Dec 12, 2012 #1
    Hi,

    1. The problem statement, all variables and given/known data

    I was hoping someone could please explain the order of variables in a Jacobian. I mean, once the dependent and independent variables have been identified, how should the Jacobian be formulated. For instance, supposing I have two implicit functions F(x,y,u,v) and G(x,y,u,v) where x and y are independent. I wish to find ∂u/∂y (x is fixed) and ∂v/∂y (x is fixed). How should the Jacobian be formulated, namely -J[(F,G)/(y,u)]/J[(F,G)/(v,u)] or -J[(F,G)/(u,y)]/J[(F,G)/(u,v)]? I happen to know the former formulation is the correct one, but why?! How could I have known and initially written it in that form?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 12, 2012 #2
    Consider two functions ##f## and ##g## of two variables y and x.
    I always form my Jacobian like
    $$
    \mathcal{J} = \begin{pmatrix}
    f_x & f_y\\
    g_x & g_y
    \end{pmatrix}
    $$
    However, I believe in some books it has my rows as columns.
     
  4. Dec 12, 2012 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    What's usually important is the absolute value of the determinant of the jacobian matrix. In which case none of these variations matter.
     
  5. Dec 12, 2012 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You can always get it from first principles (and doing this once in your life is a useful exercise). If we fix y and let x change by Δx = h, then we have:
    [tex] F(x+h,y,u + \Delta u, v + \Delta v) = 0 = F(x,y,u,v) + F_x h + F_u \Delta u + F_v \Delta v \\
    G(x+h,y,u + \Delta u, v + \Delta v) = 0 = G(x,y,u,v) + G_x h + G_u \Delta u + G_v \Delta v[/tex]
    where all the partials are evaluated at the original point (x,y,u,v). Thus,
    [tex] \pmatrix{ \Delta u \\ \Delta v} =
    - \pmatrix{F_u & F_v\\G_u&G_v}^{-1}\pmatrix{F_x \\ G_x} h,[/tex]
    so
    [tex] \pmatrix{\partial u/ \partial x \\ \partial v / \partial x} =
    - \pmatrix{F_u & F_v\\G_u&G_v}^{-1}\pmatrix{F_x \\ G_x}.[/tex]
    For a 2x2 matrix we get the inverse by swapping the diagonal elements, changing the sign of the off-diagonal elements and dividing by the determinant:
    [tex] \pmatrix{a & b \\ c & d }^{-1} = \frac{1}{ad-bc} \pmatrix{d & -b \\ -c & a},[/tex] so you can get explicit formulas for u_x and v_x.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Order of variables in a Jacobian?
Loading...